Mathematical Analysis generally, including Real Analysis, Harmonic Analysis, Complex Variable Theory, the Calculus of Variations, Measure Theory, and Non-Standard Analysis.

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0
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1answer
35 views

holomorphic function over the disk that is real on a closed curve must be constant

Let $f$ be holomorphic on $\{z\in \mathbb{C}\mid |z|\leq 3\}$ and real on the boundary of the square $\{z\in\mathbb{C}\mid Re(z)\leq1 \text{ and } Im(z)\leq 1 \}$. Prove $f$ is constant. How to ...
9
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2answers
379 views
+150

Understanding this ODE

(I did not change anything, I just rewrote the ODE in a simpler form): I started with an ODE (first ODE) : $-(1-x^2)y''(x) +x y'(x) - \left( \alpha x + \gamma x^2 \right) y(x) = \lambda y(x),$ ...
0
votes
1answer
32 views

The Landau symbol $\mathcal{o}$ as in Königsberger Analysis I

I am currently working on Chapter 14 - local approximations of function and Taylor polynomials - in Königsberger Analysis 1 Background: Königsberger introduced the Taylor Polynomial of order ...
0
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0answers
39 views

A continuous function which does not converge to its Fourier series. [duplicate]

Where can I find an example (or the theorem) for a continuous function which does not converge pointwise to its Fourier series, as well as its explanation? I would prefer a web page or a free site or ...
1
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3answers
41 views

Characterization of $\mathscr{S}(\mathbb R^n)$?

Consider the vector space $$\displaystyle\mathscr{S}(\mathbb R^n)=\{f\in C^\infty(\mathbb R^n): \lim_{|x|\to \infty} |x^\alpha \partial^\beta \phi(x)|=0, \forall \alpha, \beta\in\mathbb N_0^n\}.$$ ...
1
vote
1answer
47 views

sum of the series $\sum_{k=0}^{\infty}(k+1)(1-|x_k|)|x_n|^k$ [closed]

Let $|x_n|<1$. Is the given series convergent or which conditions do we impose on the sequence $(x_n)$ to given series be convergent? $\sum_{k=0}^{\infty}(k+1)(1-|x_k|)|x_n|^k$
1
vote
1answer
19 views

Is $(\frac{1}{1-|x_n|})$ bounded or only bounded above if $|x_n|<1$? [closed]

Let $(x_n)$ be a sequence such that $|x_n|<1$. Is $(\frac{1}{1-|x_n|})$ bounded or only bounded above if $|x_n|<1$?
1
vote
1answer
48 views

Is $(|x_n|^{n+1})$ convergent or bounded if $|x_n|<1$?

Let $(x_n)$ be a sequence such that $|x_n|<1$. Is $(|x_n|^{n+1})$ convergent or bounded?
1
vote
1answer
41 views

The existence of $f \in C^\infty(R^n)$ with $ f=0$ on closed $E$, otherwise $f>0$

This is problem 6.3 in 'Rudin's Functional analysis If $E$ is an arbitrary closed subeset of $R^n$, show that there is an $f \in C^\infty(R^n)$ such that $f(x)=0$ for every $x \in E$ and ...
0
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0answers
25 views

Taylor theorem on Sobolev spaces

I am trying to understand the Taylor theorem for Sobolev spaces that appears in http://science.org.ge/moambe/5-2/5-10%20Boyarsky.pdf. I am not sure in what sense the aproximation is. I feel that it is ...
0
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0answers
26 views

Definite integral with doable improper case

Is there a way to evaluate one or both of the following integrals: $$ \int_{a_1}^{a_2} e^{ib_1(x+b_2 \sqrt{1+x^2})}dx \quad \text{and}\quad \int_{a_1}^{a_2}\frac{x}{\sqrt{1+x^2}} e^{ib_1(x+b_2 ...
2
votes
1answer
79 views

Homework on basic inequalities.

Let $a_j$ be a sequence of positive reals. Show that (a) $\left(\sum_{j=1}^\infty a_j\right)^\theta \le \sum_{j=1}^\infty a_j^\theta$ for any $0\le\theta\le1$. (b) $\sum_{j=1}^\infty a_j^\theta \le ...
0
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1answer
26 views

How to compute the area of this set in the plane?

Let $f$ be a non-negative function which is defined, bounded, and integrable on a closed interval $[a,b]$, and let $$ S \colon= \{\ (x,y) \ | \ a \leq x \leq b, \ 0 \leq y < f(x) \ \}. $$ Then is ...
-2
votes
0answers
33 views

Does an lp norm induce a ball topology? [closed]

Namely, does the metric $$||x - y||_p$$ induce the usual ball topology that a metric induces? I wasn't able to find any results regarding this on a quick Google search.
2
votes
1answer
41 views

Unclear inequality in the proof of Birkhoff ergodic theorem.

I'm trying to understand the tricky proof of the ergodic theorem (Birkhoff 1931). My reference is "Ward,Einsiedler - Ergodic theory (with a view towards number Theory)" section 1.6: Consider the ...
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2answers
57 views

Find a power series for function

I'm having some difficulty with this problem even while noting the hint. I expressed the function as $(1/2)\frac{1}{1-(-3x/2)}$ and then thought I would work with $1/2$ of the infinite sum of ...
1
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1answer
35 views

If A has positive Haar measure then $AA^{-1}$ is a neighborhood of $e$

I read the following exercise: Prove that if $G$ is a locally compact topological group with Haar measure $\mu$ and $A \subset G, \mu (A) >0$, then $AA^{-1}$ contains an open neighborhood of the ...
1
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1answer
34 views

Having derivative in some $x_{0}$ implies having it in $U(x_{0})$ [closed]

Let $f$ be continious function in R and it has a derivative in $x_{0}$. Does it have derivative in some $U(x_{0})$?
2
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1answer
28 views

Distributions with support of the form $\left\lbrace x \right\rbrace$

Doing some calculations with Distributions I came up with the following theorem: THEOREM: Let $O \subseteq \mathbb{R^d}$ be an open subset and $x \in O$. Suppose $T \in \mathcal{D}'(O)$ with ...
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1answer
19 views

Question related to vector space of solutions of a differential equations system.

I have some doubts regarding the proof of the statement: The set of solutions of the system $$X'=A(t)X$$ where $A(t):I \subset \mathbb R \to \mathbb R^{n \times n}$ is continuous, forms an ...
0
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2answers
174 views

Why aren't all real numbers equal to one another?

I know, stupid question. But humor me for a sec. First off, we know that all real numbers have two numbers which are infinitely close to them, right? That would seem to be, for any given value of x, ...
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2answers
41 views

Can we expect, $S(\mathbb R)\ast L^{p}(\mathbb R) \subset L^{p}(\mathbb R), (1<p<\infty) $?

It is well-known that $L^{1}(\mathbb R)$ is a closed with respect to convolution(product), that is, $L^{1}(\mathbb R)\ast L^{1}(\mathbb R)\subset L^{1}(\mathbb R),$ more specifically, if $f, g\in ...
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0answers
10 views

semi linear uniform space

In semi-linear uniform space, if $f$ is a function from $(X ,Γ_X)$ to $(Y,Γ_Y)$ that is linear and bounded ,is $f$ then continuous? Is the converse true?
2
votes
1answer
46 views

The dual space of locally integrable function space

I'm strongly interested in dual spaces. I learned the dual spaces of $L^p$, $L^{\infty}$, $C(X)$ and $M(X)$. I wonder the dual space of locally integrable function space in $\mathbb{R}^n$. The ...
3
votes
2answers
51 views

$\lim_{x\to 0+}\ln(x)\cdot x = 0$ by boundedness of $\ln(x)\cdot x$

I saw a proof that $$ \lim_{x\to 0} \ln|x|\cdot x = 0 $$ where is is argued that for $x \in (0,1)$ we have $$ | \ln(x) x | = \left| \int_1^x x/t ~\mathrm d t \right| = \left| \int_x^1 x/t ~\mathrm ...
0
votes
1answer
23 views

Rewriting integrals over spheres involving $1/|x|$

The following derivation cames from calculations related to the Laplace equation and its fundamental solution. Let $g(x)$ be a test-function (meaning compact support and infinitely differentiable), ...
0
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2answers
69 views

$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$

Why does the following hold? $$\Delta_{3}\frac{1}{r}\Bigg\vert_{\mathbb{R}^3 \setminus \left\lbrace 0 \right\rbrace}=0$$
5
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2answers
98 views

Give an example of a function $f$ satisfying $\lim_{x\to 0}(f(x)f(2x))=0$,but $\lim_{x\to 0}f(x)$ does not exists

Question: Give an example of a function $f$ satisfying the condition $$\lim_{x\to 0}(f(x)f(2x))=0$$ and such that $$\lim_{x\to 0}f(x)$$ does not exists. I think this question have many example. But ...
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2answers
62 views

$f$ unbounded in $\mathbb{R}$ implies it cannot be in $L^2(\mathbb{R})$.

Intuitively , I feel this must be true. I'm looking for a rigorous proof. So,I would like to confirm that there are no counter examples to begin with.
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1answer
42 views

Questionable Intervals

Find the numbers $X_1 , X_2 , \ldots , X_{10} $ such: $X_1$ is in the interval $[0,1]$. If we divide the interval $[0,1]$ in halves,each half consists of only one of $X_1$ or $X_2$. If we divide ...
5
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1answer
63 views

$\int_0^1f(x)dx = 2, \int_0^1g(x)dx = 1, \text{and} \int_0^1[f(x)]^2 dx ≤ C$ for some constant $C > 4.$

Suppose $f$ and $g$ are nonnegative measurable functions on the interval $[0,1],$ with the properties $$\int_0^1 f(x)\,dx = 2, \int_0^1g(x)\,dx = 1, \text{ and }\int_0^1[f(x)]^2 dx \le C$$ for some ...
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0answers
52 views

sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$

Let $x_n$ be a sequence of real numbers such that $x_n\in(0,1).$ Find the sum of the series $\sum_{k=0}^{\infty}(k+1)(x_n)^k.$ My answer is $\frac{(x_n)'}{(1-x_n)^2}.$ But the term $(x_n)'$ make me ...
0
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1answer
36 views

existence of a number in (0,1) for ln(1+1/n)

Show that for each $n\in \mathbb{N}$ there is a numer $\theta_{n}$ where $0<\theta_{n}<1$ such that $log(1+ \frac{1}{n}) = \frac{1}{n} - \frac{\theta_{n}}{2n^{2}}$
0
votes
1answer
124 views

if $\mid f(x)\mid<1$ then $\mid f'(x)\mid<1$

I am wondering, if $\mid f(x)\mid<1$ then $\mid f'(x)\mid<1$. Is this true? I can not find counter example.
0
votes
1answer
47 views

Taylor's Theorem Question

Assuming Taylor's Theorem holds: Given $$F(x)=f(b)-f(x)-\frac{f'(x)}{1!}(b-x)-\cdots-\frac{f^{(n-1)}(x)}{(n-1)!}(b-x)^{n-1}$$ Show that: $$F'(x)=\frac{-f^{(n)}(x)}{n!}(b-x)^{n-1}$$
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votes
4answers
38 views

Increasing, continuous, concave downward function normalised between 0.5 and 1

What would be a good function which is increasing, continuous, concave downward with $$\lim_{x \to 0} f(x) = 0.5$$ and $$\lim_{x \to \infty} f(x) = 1.$$ It should be concave downward whose ...
1
vote
0answers
51 views

If $f_{n}\rightharpoonup \bar{f}$ and $f_{n}(x) \rightarrow f(x)$ pointwise a.e., then is $\bar{f} = f$ a.e.? [duplicate]

Suppose $f_{n}$ is a sequence of functions in $L^{p}(\mathbb{R}^{d})$ such that $\|f_{n}\|_{L^{p}} \leq 1$ for all $n$ and $f_{n}(x) \rightarrow f(x)$ pointwise almost everywhere as $n \rightarrow ...
2
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3answers
99 views

Prove that $f_n(x) → f(x)$ uniformly on $E$ as $n → ∞.$

Let $E ⊂ \mathbb{R}$ be a compact (i.e., closed bounded) set of real numbers. Suppose $\{f_n\}$ is a sequence of real-valued continuous functions which converges pointwise on $E$ to a function $f$ ...
1
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1answer
38 views

Finding rational points at rational distance in the plane

Take any point $p$ in the real plane. Does there always exist a rational point at a rational distance from $p$? (A rational point is a point $(q,r)$ where $q$ and $r$ are rational.)
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2answers
25 views

Show that $\int_X gdν=\int_X gfdμ$ for all $g∈L_1(ν).$

Let $μ$ and $ν$ be finite (positive) measures on a measurable space $(X, M),$ and suppose that $ν(E)=\int_E fdμ$, for all $E∈M,$ $E$ where $f$ is some function in $L_1(μ).$ Show that $\int_X ...
0
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1answer
24 views

Lipschitz function proof

Statement Let $F(t,X)=A(t)X+b(t)$ with $A(t) \in \mathbb R^{n\times n}$ and $b(t) \in \mathbb R^n$. If the coefficients $a_{ij}(t)$ and $b_i(t)$ are continuous functions of the variable $t$ in a ...
0
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0answers
12 views

Estimate of Projection Operator on two-torus

Let $\Lambda$ be a lattice, $\mathbb{T}=\mathbb{R}^2/\Lambda$ be a flat torus and $\Delta$ be the Laplace-Beltrami operator. There is any reference where the norm of the projection operator ...
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3answers
36 views

Computation of surfaces areas of some objects

I want to calculate the surface area of the following objects: 1) A cylinder with height $h$ and radius $r$ 2) A cone $C=\{(x,y,z) \in \mathbb R^3 : x^2+y^2=z^2, 0<z<4\}$ 3) A torus At first ...
2
votes
1answer
58 views

Show solvability of ODE without explicitly calculating solution

Show that $$ u + u^{(4)} - u^{(2)} = f $$ has a solution $u \in H^4(\mathbb R)$ (without explicitly calculuting it) for every $f \in L^2(\mathbb R)$! What criteria for solvability for such ODE's ...
4
votes
1answer
110 views

How to integrate these integrals? $\int{\frac{1}{(x^x-x^{-x})}} dx$ [closed]

Problem : $$\int{\frac{1}{(x^x-x^{-x})}} dx$$ I need answer about the following problem. Please help . I will be grateful to you. Thanks.
3
votes
1answer
25 views

Triangle inequality and homomorphisms

Here is my situation: I have two homomorphisms $f$ and $g$ from a group $A$ into the complex numbers $\mathbb{C}$. I know that they are 'close' on a subset $B \subseteq A$. More formally there is an ...
4
votes
1answer
33 views

Locally integrable function with a uniform bound…

I'm a bit lost... I have a measure space $(\Omega,\mathcal{B}(\Omega),\mu)$ where $\mathcal{B}(\Omega)$ is a Borel set. Let $f$ be a real-valued measurable function on $\Omega$ and $\mathcal{K}$ be ...
1
vote
1answer
24 views

Difference in Definitions of Quasiconvexity

So I've seen a few different definitions of quasiconvexity of a function and I cannot, after a bit of working, figure out how all of them are related: Let $X$ be a convex subset of a real vector ...
1
vote
1answer
37 views

nth term derivative with f(0) plugged in

Question: compute the sixth derivative of $\frac{\cos{(5x^2)}-1}{x^2}$ and plug in zero to the derivative. What is the answer? I keep concluding that this question should be 892857 when I plug in 0 ...
4
votes
1answer
68 views

Riemann Sums as in Königsberger Analysis 1

Intro: I must take a small detour here which is only relevant if you do not know the book itself and care about my background. I am working with Königsberger Analysis I (can be found on Springerlink). ...